Optimal. Leaf size=24 \[ x^2 \left (-1+e+\frac {3 x}{2}+\frac {1}{x \log \left (x^2\right )}\right )^2 \]
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Rubi [B] time = 0.30, antiderivative size = 55, normalized size of antiderivative = 2.29, number of steps used = 21, number of rules used = 10, integrand size = 94, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.106, Rules used = {6688, 14, 2302, 30, 2320, 2330, 2300, 2178, 2307, 2298} \begin {gather*} \frac {9 x^4}{4}-3 (1-e) x^3+(1-e)^2 x^2+\frac {1}{\log ^2\left (x^2\right )}-\frac {(2 (1-e)-3 x) x}{\log \left (x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 2178
Rule 2298
Rule 2300
Rule 2302
Rule 2307
Rule 2320
Rule 2330
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (x \left (2 (1-e)^2-9 (1-e) x+9 x^2\right )-\frac {4}{x \log ^3\left (x^2\right )}+\frac {4-4 e-6 x}{\log ^2\left (x^2\right )}+\frac {2 (-1+e+3 x)}{\log \left (x^2\right )}\right ) \, dx\\ &=2 \int \frac {-1+e+3 x}{\log \left (x^2\right )} \, dx-4 \int \frac {1}{x \log ^3\left (x^2\right )} \, dx+\int x \left (2 (1-e)^2-9 (1-e) x+9 x^2\right ) \, dx+\int \frac {4-4 e-6 x}{\log ^2\left (x^2\right )} \, dx\\ &=-\frac {(2 (1-e)-3 x) x}{\log \left (x^2\right )}+2 \int \left (\frac {-1+e}{\log \left (x^2\right )}+\frac {3 x}{\log \left (x^2\right )}\right ) \, dx-2 \operatorname {Subst}\left (\int \frac {1}{x^3} \, dx,x,\log \left (x^2\right )\right )-(2 (1-e)) \int \frac {1}{\log \left (x^2\right )} \, dx+\int \left (2 (-1+e)^2 x+9 (-1+e) x^2+9 x^3\right ) \, dx+\int \frac {4-4 e-6 x}{\log \left (x^2\right )} \, dx\\ &=(1-e)^2 x^2-3 (1-e) x^3+\frac {9 x^4}{4}+\frac {1}{\log ^2\left (x^2\right )}-\frac {(2 (1-e)-3 x) x}{\log \left (x^2\right )}+6 \int \frac {x}{\log \left (x^2\right )} \, dx-(2 (1-e)) \int \frac {1}{\log \left (x^2\right )} \, dx-\frac {((1-e) x) \operatorname {Subst}\left (\int \frac {e^{x/2}}{x} \, dx,x,\log \left (x^2\right )\right )}{\sqrt {x^2}}+\int \left (\frac {4 (1-e)}{\log \left (x^2\right )}-\frac {6 x}{\log \left (x^2\right )}\right ) \, dx\\ &=(1-e)^2 x^2-3 (1-e) x^3+\frac {9 x^4}{4}-\frac {(1-e) x \text {Ei}\left (\frac {\log \left (x^2\right )}{2}\right )}{\sqrt {x^2}}+\frac {1}{\log ^2\left (x^2\right )}-\frac {(2 (1-e)-3 x) x}{\log \left (x^2\right )}+3 \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,x^2\right )-6 \int \frac {x}{\log \left (x^2\right )} \, dx+(4 (1-e)) \int \frac {1}{\log \left (x^2\right )} \, dx-\frac {((1-e) x) \operatorname {Subst}\left (\int \frac {e^{x/2}}{x} \, dx,x,\log \left (x^2\right )\right )}{\sqrt {x^2}}\\ &=(1-e)^2 x^2-3 (1-e) x^3+\frac {9 x^4}{4}-\frac {2 (1-e) x \text {Ei}\left (\frac {\log \left (x^2\right )}{2}\right )}{\sqrt {x^2}}+\frac {1}{\log ^2\left (x^2\right )}-\frac {(2 (1-e)-3 x) x}{\log \left (x^2\right )}+3 \text {li}\left (x^2\right )-3 \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,x^2\right )+\frac {(2 (1-e) x) \operatorname {Subst}\left (\int \frac {e^{x/2}}{x} \, dx,x,\log \left (x^2\right )\right )}{\sqrt {x^2}}\\ &=(1-e)^2 x^2-3 (1-e) x^3+\frac {9 x^4}{4}+\frac {1}{\log ^2\left (x^2\right )}-\frac {(2 (1-e)-3 x) x}{\log \left (x^2\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.07, size = 28, normalized size = 1.17 \begin {gather*} \frac {\left (2+x (-2+2 e+3 x) \log \left (x^2\right )\right )^2}{4 \log ^2\left (x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 75, normalized size = 3.12 \begin {gather*} \frac {{\left (9 \, x^{4} - 12 \, x^{3} + 4 \, x^{2} e^{2} + 4 \, x^{2} + 4 \, {\left (3 \, x^{3} - 2 \, x^{2}\right )} e\right )} \log \left (x^{2}\right )^{2} + 4 \, {\left (3 \, x^{2} + 2 \, x e - 2 \, x\right )} \log \left (x^{2}\right ) + 4}{4 \, \log \left (x^{2}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 107, normalized size = 4.46 \begin {gather*} \frac {9 \, x^{4} \log \left (x^{2}\right )^{2} + 12 \, x^{3} e \log \left (x^{2}\right )^{2} - 12 \, x^{3} \log \left (x^{2}\right )^{2} + 4 \, x^{2} e^{2} \log \left (x^{2}\right )^{2} - 8 \, x^{2} e \log \left (x^{2}\right )^{2} + 4 \, x^{2} \log \left (x^{2}\right )^{2} + 12 \, x^{2} \log \left (x^{2}\right ) + 8 \, x e \log \left (x^{2}\right ) - 8 \, x \log \left (x^{2}\right ) + 4}{4 \, \log \left (x^{2}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 69, normalized size = 2.88
method | result | size |
risch | \(\frac {9 x^{4}}{4}+3 x^{3} {\mathrm e}-3 x^{3}+x^{2} {\mathrm e}^{2}-2 x^{2} {\mathrm e}+x^{2}+\frac {2 x \,{\mathrm e} \ln \left (x^{2}\right )+3 x^{2} \ln \left (x^{2}\right )-2 x \ln \left (x^{2}\right )+1}{\ln \left (x^{2}\right )^{2}}\) | \(69\) |
norman | \(\frac {1+\left (-3+3 \,{\mathrm e}\right ) x^{3} \ln \left (x^{2}\right )^{2}+\left (2 \,{\mathrm e}-2\right ) x \ln \left (x^{2}\right )+\left ({\mathrm e}^{2}-2 \,{\mathrm e}+1\right ) x^{2} \ln \left (x^{2}\right )^{2}+3 x^{2} \ln \left (x^{2}\right )+\frac {9 x^{4} \ln \left (x^{2}\right )^{2}}{4}}{\ln \left (x^{2}\right )^{2}}\) | \(78\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 59, normalized size = 2.46 \begin {gather*} \frac {9}{4} \, x^{4} + 3 \, x^{3} e - 3 \, x^{3} + x^{2} e^{2} - 2 \, x^{2} e + x^{2} + \frac {3 \, x^{2} + 2 \, x {\left (e - 1\right )}}{2 \, \log \relax (x)} + \frac {1}{\log \left (x^{2}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.29, size = 47, normalized size = 1.96 \begin {gather*} \frac {{\left (2\,x\,\mathrm {e}-2\,x+3\,x^2\right )}^2}{4}+\frac {\ln \left (x^2\right )\,\left (2\,x\,\mathrm {e}-2\,x+3\,x^2\right )+1}{{\ln \left (x^2\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.15, size = 58, normalized size = 2.42 \begin {gather*} \frac {9 x^{4}}{4} + x^{3} \left (-3 + 3 e\right ) + x^{2} \left (- 2 e + 1 + e^{2}\right ) + \frac {\left (3 x^{2} - 2 x + 2 e x\right ) \log {\left (x^{2} \right )} + 1}{\log {\left (x^{2} \right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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